The necessity of bounding the next term will become evident in eq.Rn,eq.Rn1,eq.Rn2,eq.da-Rn2. For \({1\le r\le m \le h}\) , the chain rule for powers of functions (see for example [1]}) and our derived estimates eq.vn-1,eq.d1-vn,eq.d2-vn,eq.dl-vn gives
\({\Big |\frac{1}{h!}{\frac{\mathrm {d}^{r}}{\mathrm {d}t^{r}}}v_n^h(t,z)\Big |=\Big |\frac{r!\cdot h!}{h!} \sum _{u=1}^{r\wedge h} \sum _{*(k_\cdot ,r,u)} \frac{v_n^{h-u}(t,z)}{(h-u)!} \prod _{l=1}^{r} \frac{1}{k_l!} \Big (\frac{1}{l!}{\frac{\mathrm {d}^{l}}{\mathrm {d}t^{l}}}v_n(t,z)\Big )^{k_l}\Big |}\qquad \\&\le c(m)\sum _{u=1}^{r} \sum _{*(k_\cdot ,r,u)} \frac{|v_n(t,z)|^{h-m}}{(h-m)!}|v_n(t,z)|^{m-u} \prod _{l=1}^{r} \Big |{\frac{\mathrm {d}^{l}}{\mathrm {d}t^{l}}}v_n(t,z)\Big |^{k_l}\\&\le c(m) \frac{|v_n(t,z)|^{h-m}}{(h-m)!} \sum _{u=1}^{r} \sum _{*(k_\cdot ,r,u)}\big (|t|^3\widetilde{L}_{3,n}\big )^{m-u}\big (|t|^2\widetilde{L}_{3,n}\big )^{k_1}\big (|t|\widetilde{L}_{3,n}\big )^{k_2}\prod _{l=3}^{r} \widetilde{L}_{l,n}^{k_l}\\&= c(m) \frac{|v_n(t,z)|^{h-m}}{(h-m)!} \sum _{u=1}^{r} \sum _{*(k_\cdot ,r,u)}|t|^{3m-3u+2k_1+k_2}\widetilde{L}_{3,n}^{m-u+k_1+k_2}\prod _{l=3}^{r} \widetilde{L}_{l,n}^{k_l}.\) 
